Even in that context, it doesn't make much sense for Joel or Tommy to offer up their real names to complete strangers. It would have worked a lot better if they had both given Abby fake names, and then in the course of events Tommy had accidentally used the name "Joel."
There's literally never been a single instance of any character in this series giving out a fake name. Because they genuinely have no reason to do so. Social media is dead. The only way anybody finds out about anybody is hearsay, and in the most extreme circumstances people lose their individuality entirely if they're apart of a well known group. Take the fireflies for example. No one ever perceives them by their names, or relationships. It's always. "You're part of the fireflies." With the only way to know about anybody else is vague conversations. Joel wasn't some legend. Everyone he encounters is by coincidence.